Golden ratio

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Golden ratio

In mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
and the art

Art

Art is the process or product of deliberately arranging elements in a way that appeals to the senses or emotions. It encompasses a diverse range of human activities, creations, and modes of expression, including music and literature....
s, two quantities are in the golden ratio

Golden ratio

In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller....
if the ratio

Ratio

A ratio is an expression which compares quantities relative to each other. The most common examples involve two quantities, but in theory any number of quantities can be compared....
between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller. The golden ratio is an irrational

Irrational number

Mathematical constant

A mathematical constant is a number, usually a real number, that arises naturally in mathematics. Unlike physical constants, mathematical constants are defined independently of physical measurement....
, approximately 1.6180339887.

At least since the Renaissance

Renaissance

The Renaissance was a cultural movement that spanned roughly the 14th to the 17th century, beginning in Italy in the late Middle Ages and later spreading to the rest of Europe....
, many artist

Artist

The definition of an artist is wide-ranging and covers a broad spectrum of activities to do with creating art, practicing the arts and/or demonstrating an art....
s and architect

Architect

An architect is trained and licenced in planning and designing buildings, and participates in supervising the construction of a building. Etymologically, architect derives from the Latin architectus, itself derived from the Greek arkhitekton , i.e....
s have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle

Golden rectangle

A golden rectangle is a rectangle whose side lengths are in the golden ratio, 1: , that is, or approximately 1:1.618.A distinctive feature of this shape is that when a square section is removed, the remainder is another golden rectangle; that is, with the same proportionality s as the first....
, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically

Aesthetics

Aesthetics or esthetics is commonly known as the study of senses or sensori-emotional values, sometimes called judgments of sentiment and taste ....
pleasing.

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Encyclopedia

In mathematics

Mathematics

Art

Golden ratio

Ratio

Irrational number

Mathematical constant

Renaissance

The Renaissance was a cultural movement that spanned roughly the 14th to the 17th century, beginning in Italy in the late Middle Ages and later spreading to the rest of Europe....
, many artist

Artist

The definition of an artist is wide-ranging and covers a broad spectrum of activities to do with creating art, practicing the arts and/or demonstrating an art....
s and architect

Architect

Golden rectangle

Aesthetics

Aesthetics or esthetics is commonly known as the study of senses or sensori-emotional values, sometimes called judgments of sentiment and taste ....
pleasing. Mathematician

Mathematician

A mathematician is a person whose primary area of study and/or research is the field of mathematics....
s have studied the golden ratio because of its unique and interesting properties.

The golden ratio is often denoted by the Greek

Greek alphabet

The Greek alphabet is a set of twenty-four letters that has been used to write the Greek language since the late 9th century BC or early 8th century BCE....
letter F (phi

Phi (letter)

Phi , pronounced [] in Modern Greek language and as [] in English, is the 21st letter of the Greek alphabet. In modern Greek, it represents [], a voiceless labiodental fricative....
). The figure of a golden section illustrates the geometric relationship that defines this constant. Expressed algebraically:

This equation has as its unique positive solution the algebraic

Algebraic number

In mathematics, an algebraic number is a complex number that is a root of a non-zero polynomial in one variable with rational number coefficients....
irrational number

Irrational number

In mathematics, an irrational number is any real number that is not a rational number ? that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero....

Other names frequently used for or closely related to the golden ratio are golden section (Latin: sectio aurea), golden mean, golden number, and the Greek letter phi
Phi (letter)

Phi , pronounced [] in Modern Greek language and as [] in English, is the 21st letter of the Greek alphabet. In modern Greek, it represents [], a voiceless labiodental fricative....
(F). Other terms encountered include extreme and mean ratio, medial section, divine proportion, divine section (Latin: sectio divina), golden proportion, golden cut, and mean of Phidias
Phidias

Phidias or Pheidias; ; circa 480 BC 430 BC), was a Hellenic civilization sculptor, painter and architect, who lived in the Classical Greece, in the 5th century BC, and is commonly regarded as one of the greatest of all Classical sculptors....
.

Calculation

Two quantities (positive numbers) a and b are said to be in the golden ratio ? if:

This equation unambiguously defines ?.

The right equation shows that a = b?, which can be substituted in the left part, giving

Dividing out b yields

Multiplying both sides by ? and rearranging terms leads to:

The only positive solution to this quadratic equation

Quadratic equation

In mathematics, a quadratic equation is a polynomial equation of the second degree of a polynomial. The general form iswhere a ? 0. The letters a, b, and c are called coefficients: the quadratic coefficient a is the coefficient of x2, the linear coefficient b is the coefficient of x, and c i...
is

History

The golden ratio has fascinated Western intellectuals of diverse interests for at least 2,400 years:

Ancient Greek

Ancient Greece

The term Ancient Greece refers to the period of History of Greece lasting from the Greek Dark Ages ca. 1100 BC and the Dorian invasion, to 146 BC and the Roman Republic conquest of Greece after the Battle of Corinth ....
mathematicians first studied what we now call the golden ratio because of its frequent appearance in geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
. The division of a line into "extreme and mean ratio" (the golden section) is important in the geometry of regular pentagram

Pentagram

Pentagon

In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540?....
s. The Greeks usually attributed discovery of this concept to Pythagoras

Pythagoras

Pythagoras of Samos was an Ionians Ancient Greeks mathematician and founder of the religious movement called Pythagoreanism. He is often revered as a great mathematician, mysticism and scientist; however some have questioned the scope of his contributions to mathematics and natural philosophy....
or his followers

Pythagoreanism

Pythagoreanism is a term used for the esoteric and metaphysics beliefs held by Pythagoras and his followers, the Pythagoreans, who were much influenced by mathematics and probably a very inspirational source for Plato and Platonism....
. The regular pentagram, which has a regular pentagon inscribed within it, was the Pythagoreans' symbol.

Euclid

Euclid

Euclid , floruit 300 BC, also known as Euclid of Alexandria, was a Greek mathematics and is often referred to as the Father of Geometry. He was active in Alexandria during the reign of Ptolemy I ....
's Elements

Euclid's Elements

Euclid's Elements is a mathematics and geometry treatise consisting of 13 books written by the Greek mathematics Euclid in Alexandria circa 300 BC....
(Greek

Greek language

Greek is an Indo-European languages native to the southern Balkan peninsula, the language of the Greek people. It forms an independent branch within Indo-European....
: ) provides the first known written definition of what is now called the golden ratio: "A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less." Euclid explains a construction for cutting (sectioning) a line "in extreme and mean ratio", i.e. the golden ratio. Throughout the Elements, several propositions (theorem

Theorem

In mathematics, a theorem is a statement Mathematical proof on the basis of previously accepted or established statements such as axioms.In formal mathematical logic, the concept of a theorem may be taken to mean a formula that can be formal proof according to the deductive system of a fixed formal system....
s in modern terminology) and their proofs employ the golden ratio. Some of these propositions show that the golden ratio is an irrational number

Irrational number

In mathematics, an irrational number is any real number that is not a rational number ? that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero....
.

The name "extreme and mean ratio" was the principal term used from the 3rd century BC until about the 18th century.

The modern history of the golden ratio starts with Luca Pacioli

Luca Pacioli

Fra Luca Bartolomeo de Pacioli was an Italy mathematician and Franciscan friar, collaborator with Leonardo da Vinci, and seminal contributor to the field now known as accounting....
's Divina Proportione of 1509, which captured the imagination of artists, architects, scientists, and mystics with the properties, mathematical and otherwise, of the golden ratio.

The first known approximation of the (inverse) golden ratio by a decimal fraction, stated as "about 0.6180340," was written in 1597 by Prof. Michael Maestlin

Michael Maestlin

Michael Maestlin was a Germany astronomer and mathematician, known for being the mentor of Johannes Kepler....
of the University of Tübingen in a letter to his former student Johannes Kepler

Johannes Kepler

Johannes Kepler was a Germans mathematician, astronomer and astrologer, and key figure in the 17th century Scientific revolution. He is best known for his eponymous Kepler's laws of planetary motion, codified by later astronomers based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican Astrononomy....
.

Since the twentieth century, the golden ratio has been represented by the Greek letter

Greek alphabet

The Greek alphabet is a set of twenty-four letters that has been used to write the Greek language since the late 9th century BC or early 8th century BCE....
? (phi

PHI

PHI is a three-letter acronym or abbreviation that can refer to:* Post-Polio Health International* Protected Health Information as part of the HIPAA regulations...
, after Phidias

Phidias

Phidias or Pheidias; ; circa 480 BC 430 BC), was a Hellenic civilization sculptor, painter and architect, who lived in the Classical Greece, in the 5th century BC, and is commonly regarded as one of the greatest of all Classical sculptors....
, a sculptor who is said to have employed it) or less commonly by t (tau

Tau

Tau is the 19th letter of the Greek alphabet. In the system of Greek numerals it has a value of 300. This letter in English is pronounced , but in Modern Greek, this letter's name is pronounced ....
, the first letter of the ancient Greek

Ancient Greek

Ancient Greek is the historical stage in the development of the Greek language spanning across the Archaic Greece , Classical Greece , and Hellenistic civilization periods of ancient Greece and the classical antiquity....
root t?µ?—meaning cut).

Timeline

Timeline according to Priya Hemenway.

Phidias
Phidias

Phidias or Pheidias; ; circa 480 BC 430 BC), was a Hellenic civilization sculptor, painter and architect, who lived in the Classical Greece, in the 5th century BC, and is commonly regarded as one of the greatest of all Classical sculptors....
(490–430 BC) made the Parthenon
Parthenon

The Parthenon is a Greek temple of the Greek gods Athena, built in the 5th century BC on the Acropolis of Athens. It is the most important surviving building of Classical Greece, generally considered to be the culmination of the development of the Doric order....
statues that seem to embody the golden ratio.
Plato
Plato

Plato , was a Classical Greece Greeks philosopher, mathematician, writer of philosophical dialogues, and founder of the Platonic Academy in Ancient Athens, the first institution of higher learning in the western world....
(427–347 BC), in his Timaeus
Timaeus

Timaeus is a Greek name, meaning "Honour". It may refer to:*Timaeus , a Socratic dialogue by Plato*Timaeus of Locri, the 5th-century Pythagorean philosopher, appearing in Plato's dialogue...
, describes five possible regular solids (the Platonic solids, the tetrahedron
Tetrahedron

A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....
, cube
Cube

A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
, octahedron
Octahedron

An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....
, dodecahedron
Dodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
and icosahedron
Icosahedron

In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....
), some of which are related to the golden ratio.
Euclid
Euclid

Euclid , floruit 300 BC, also known as Euclid of Alexandria, was a Greek mathematics and is often referred to as the Father of Geometry. He was active in Alexandria during the reign of Ptolemy I ....
(c. 325–c. 265 BC), in his Elements
Euclid's Elements

Euclid's Elements is a mathematics and geometry treatise consisting of 13 books written by the Greek mathematics Euclid in Alexandria circa 300 BC....
, gave the first recorded definition of the golden ratio, which he called, as translated into English, "extreme and mean ratio" (Greek: a???? ?a? µes?? ?????).
Fibonacci
Fibonacci

Leonardo of Pisa , also known as Leonardo Pisano, Leonardo Bonacci, Leonardo Fibonacci, or, most commonly, simply Fibonacci, was an Italy mathematician, considered by some "the most talented mathematician of the Middle Ages"....
(1170–1250) mentioned the numerical series
Sequence

In mathematics, a sequence is an ordered list of objects . Like a Set , it contains Element , and the number of terms is called the length of the sequence....
now named after him in his Liber Abaci
Liber Abaci

Liber Abaci is a historic book on arithmetic by Leonardo of Pisa, known later by his nickname Fibonacci. Its title has two common translations, The Book of the Abacus or The Book of Calculation....
; the Fibonacci sequence
Fibonacci number

In mathematics, the Fibonacci numbers are a sequence of numbers named after Leonardo of Pisa, known as Fibonacci . Fibonacci's 1202 book Liber Abaci introduced the sequence to Western European mathematics, although the sequence had been previously described in Indian mathematics....
is closely related to the golden ratio.
Luca Pacioli
Luca Pacioli

Fra Luca Bartolomeo de Pacioli was an Italy mathematician and Franciscan friar, collaborator with Leonardo da Vinci, and seminal contributor to the field now known as accounting....
(1445–1517) defines the golden ratio as the "divine proportion" in his Divina Proportione.
Johannes Kepler
Johannes Kepler

Johannes Kepler was a Germans mathematician, astronomer and astrologer, and key figure in the 17th century Scientific revolution. He is best known for his eponymous Kepler's laws of planetary motion, codified by later astronomers based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican Astrononomy....
(1571–1630) describes the golden ratio as a "precious jewel": "Geometry has two great treasures: one is the Theorem of Pythagoras
Pythagorean theorem

In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a triangle#Types of triangles....
, and the other the division of a line into extreme and mean ratio; the first we may compare to a measure of gold, the second we may name a precious jewel." These two treasures are combined in the Kepler triangle
Kepler triangle

A Kepler triangle is a Special right triangle with edge lengths in geometric progression. The ratio of the edges of a Kepler triangle are linked to the golden ratio...
.
Charles Bonnet
Charles Bonnet

Charles Bonnet , Switzerland natural history and philosophical writer, was born at Geneva, of a France family driven into Switzerland by the religious persecution in the 16th century....
(1720–1793) points out that in the spiral phyllotaxis
Phyllotaxis

In botany, phyllotaxis or phyllotaxy is the arrangement of the leaf on the plant stem of a plant....
of plants going clockwise
Clockwise

A clockwise motion is one that proceeds 'like the clock's hands': from the top to the right, then down and then to the left, and back to the top....
and counter-clockwise were frequently two successive Fibonacci series.
Martin Ohm
Martin Ohm

Martin Ohm was a Germany mathematician and a younger brother of physicist Georg Ohm. He earned his doctorate in 1811 at Friedrich-Alexander-University, Erlangen-Nuremberg where his advisor was Karl Christian von Langsdorf....
(1792–1872) is believed to be the first to use the term goldener Schnitt (golden section) to describe this ratio, in 1835.

Edouard Lucas
Edouard Lucas

Fran?ois ?douard Anatole Lucas was a France mathematician. Lucas is known for his study of the Fibonacci sequence. The related Lucas sequence is named after him....
(1842–1891) gives the numerical sequence now known as the Fibonacci sequence its present name.
Mark Barr
Mark Barr

Mark Barr was an American mathematician who, according to Theodore Andrea Cook, in about 1909, gave the golden ratio the name of phi, the first Greek letter in the name of Phidias, the Greek sculptor who lived around 450 BC....
(20th century) suggests the Greek letter phi (f), the initial letter of Greek sculptor Phidias's name, as a symbol
Symbol

A symbol is something such as an entity, picture, written word, sound, or particular mark that represents something else by association, resemblance, or convention....
for the golden ratio.
Roger Penrose
Roger Penrose

Sir Roger Penrose, Order of Merit , Royal Society is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College....
(b.1931) discovered a symmetrical pattern that uses the golden ratio in the field of aperiodic tiling
Aperiodic tiling

The informal term aperiodic tiling loosely refers to an aperiodic set of tiles and the tilings which such sets admit. Properly speaking, aperiodicity is a property of the set of tiles themselves; a given tiling is simply non-periodic or periodic....
s, which led to new discoveries about quasicrystals.

Aesthetics

Beginning in the Renaissance

Renaissance

The Renaissance was a cultural movement that spanned roughly the 14th to the 17th century, beginning in Italy in the late Middle Ages and later spreading to the rest of Europe....
, a body of literature on the aesthetics

Aesthetics

Aesthetics or esthetics is commonly known as the study of senses or sensori-emotional values, sometimes called judgments of sentiment and taste ....
of the golden ratio has developed. As a result, architects, artists, book designers, and others have been encouraged to use the golden ratio in the dimensional relationships of their works.

The first and most influential of these was De Divina Proportione by Luca Pacioli

Luca Pacioli

Franciscan

The term Franciscan is commonly used to refer to members of Catholic religious orders that follow a body of regulations known as "The rule of St....
friar

Friar

A friar is a member of one of the mendicant orders....
, was known mostly as a mathematician, but he was also trained and keenly interested in art. De Divina Proportione explored the mathematics of the golden ratio. Though it is often said that Pacioli advocated the golden ratio's application to yield pleasing, harmonious proportions, Livio points out that that interpretation has been traced to an error in 1799, and that Pacioli actually advocated the Vitruvian

Vitruvius

File:Vitruvius.jpgMarcus Vitruvius Pollio was a Ancient Rome writer, architect and engineer , active in the 1st century BC. By his own description Vitruvius served as a Ballista , the third class of arms in the military offices....
system of rational proportions. Pacioli also saw Catholic religious significance in the ratio, which led to his work's title. Containing illustrations of regular solids by Leonardo Da Vinci

Leonardo da Vinci

Leonardo di ser Piero da Vinci was an Italy polymath, being a scientist, mathematician, engineer, inventor, anatomist, Painting, sculptor, architect, botanist, musician and writer....
, Pacioli's longtime friend and collaborator, De Divina Proportione was a major influence on generations of artists and architects alike.

Architecture

Some studies of the Acropolis

Acropolis

Acropolis literally means city on the edge . For purposes of defense, early settlers naturally chose elevated ground, frequently a hill with precipitous sides....
, including the Parthenon

Parthenon

The Parthenon is a Greek temple of the Greek gods Athena, built in the 5th century BC on the Acropolis of Athens. It is the most important surviving building of Classical Greece, generally considered to be the culmination of the development of the Doric order....
, conclude that many of its proportions approximate the golden ratio. The Parthenon's facade as well as elements of its facade and elsewhere can be circumscribed by golden rectangles. To the extent that classical buildings or their elements are proportioned according to the golden ratio, this might indicate that their architects were aware of the golden ratio and consciously employed it in their designs. Alternatively, it is possible that the architects used their own sense of good proportion, and that this led to some proportions that closely approximate the golden ratio. On the other hand, such retrospective analyses can always be questioned on the ground that the investigator chooses the points from which measurements are made or where to superimpose golden rectangles, and that these choices affect the proportions observed.

Some scholars deny that the Greeks had any aesthetic association with golden ratio. For example, Midhat J. Gazalé says, "It was not until Euclid, however, that the golden ratio's mathematical properties were studied. In the Elements (308 B.C.) the Greek mathematician merely regarded that number as an interesting irrational number, in connection with the middle and extreme ratios. Its occurrence in regular pentagons and decagons was duly observed, as well as in the dodecahedron (a regular polyhedron

Regular polyhedron

A regular polyhedron is a polyhedron whose faces are Congruence regular polygons which are assembled in the same way around each vertex. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive - i.e....
whose twelve faces are regular pentagons). It is indeed exemplary that the great Euclid, contrary to generations of mystics who followed, would soberly treat that number for what it is, without attaching to it other than its factual properties." And Keith Devlin

Keith Devlin

Keith J. Devlin is an England mathematician and writer....
says, "Certainly, the oft repeated assertion that the Parthenon in Athens is based on the golden ratio is not supported by actual measurements. In fact, the entire story about the Greeks and golden ratio seems to be without foundation. The one thing we know for sure is that Euclid, in his famous textbook Elements, written around 300 B.C., showed how to calculate its value." Near-contemporary sources like Vitruvius

Vitruvius

File:Vitruvius.jpgMarcus Vitruvius Pollio was a Ancient Rome writer, architect and engineer , active in the 1st century BC. By his own description Vitruvius served as a Ballista , the third class of arms in the military offices....
exclusively discuss proportions that can be expressed in whole numbers, i.e. commensurate as opposed to irrational proportions.

A geometrical analysis of the Great Mosque of Kairouan

Mosque of Oqba

The Mosque of Uqba , also known as the Great Mosque of Kairouan , is one of the most important mosques in Tunisia. Seven pilgrimages to this mosque is considered the equivalent of one pilgrimage to Mecca....
reveals a consistent application of the golden ratio throughout the design, according to Boussora and Mazouz. It is found in the overall proportion of the plan and in the dimensioning of the prayer space, the court, and the minaret

Minaret

Minarets are distinctive architectural features of Islamic mosques. Minarets are generally tall spires with onion dome, usually either free standing or much taller than any surrounding support structure....
. Boussora and Mazouz also examined earlier archaeological theories about the mosque, and demonstrate the geometric constructions based on the golden ratio by applying these constructions to the plan of the mosque to test their hypothesis.

The Swiss architect

Architect

Le Corbusier

Charles-?douard Jeanneret-Gris, who chose to be known as Le Corbusier , was a Swiss-French architect, designer, urbanist, writer and also Painting, who is famous for being one of the pioneers of what now is called Modern architecture or the International Style....
, famous for his contributions to the modern

Modernism

Modernism, in its broadest definition, is modern thought, character, or practice. More specifically, the term describes both a set of cultural tendencies and an array of associated cultural movements, originally arising from wide-scale and far-reaching changes to Western culture in the late nineteenth and early twentieth century....
international style

International style (architecture)

The International style was a major architectural style of the 1920s and 1930s. The term usually refers to the buildings and architects of the formative decades of Modernism, before World War II....
, centered his design philosophy on systems of harmony and proportion. Le Corbusier's faith in the mathematical order of the universe was closely bound to the golden ratio and the Fibonacci series, which he described as "rhythms apparent to the eye and clear in their relations with one another. And these rhythms are at the very root of human activities. They resound in man by an organic inevitability, the same fine inevitability which causes the tracing out of the Golden Section by children, old men, savages and the learned."

Le Corbusier explicitly used the golden ratio in his Modulor

Modulor

The Modulor is a scale of Proportion s devised by the List of French people, List of Swiss people architect Le Corbusier ....
system for the scale

Scale (ratio)

The concept of scale is applicable if a system is represented Proportionality ly by another system. For example, for a scale model of an object, the ratio of corresponding lengths is a Dimensionless number scale, e.g....
of architectural proportion

Proportion (architecture)

Proportion is the relation between elements and a whole....
. He saw this system as a continuation of the long tradition of Vitruvius

Vitruvius

Vitruvian Man

The Vitruvian Man is a world-renowned drawing with accompanying notes created by Leonardo da Vinci around the year 1487 as recorded in one of his journals....
", the work of Leon Battista Alberti, and others who used the proportions of the human body to improve the appearance and function of architecture

Architecture

The term architecture can refer to a process, a profession or documentation.As a process, architecture is the activity of designing and construction buildings and other physical structures by a person or a computer, primarily to provide shelter....
. In addition to the golden ratio, Le Corbusier based the system on human measurements

Anthropometry

Anthropometry , in physical anthropology, refers to the measurement of the human individual for the purposes of understanding human physical variation....
, Fibonacci numbers, and the double unit. He took Leonardo's suggestion of the golden ratio in human proportions to an extreme: he sectioned his model human body's height at the navel with the two sections in golden ratio, then subdivided those sections in golden ratio at the knees and throat; he used these golden ratio proportions in the Modulor

Modulor

The Modulor is a scale of Proportion s devised by the List of French people, List of Swiss people architect Le Corbusier ....
system. Le Corbusier's 1927 Villa Stein in Garches

Garches

Garches is a commune in France in the western suburbs of Paris, France. It is located . from the Kilometre Zero....
exemplified the Modulor system's application. The villa's rectangular ground plan, elevation, and inner structure closely approximate golden rectangles.

Another Swiss architect, Mario Botta

Mario Botta

Mario Botta is a famous modern architect born in Mendrisio, Ticino canton, Switzerland.He designed his first house at age 16, although no-one mentions if it was built, and studied at the Liceo Artistico in Milan and the University Iuav of Venice in Venice....
, bases many of his designs on geometric figures. Several private houses he designed in Switzerland are composed of squares and circles, cubes and cylinders. In a house he designed in Origlio

Origlio

Origlio is a Municipalities of Switzerland in the district of Lugano in the Cantons of Switzerland of Ticino in Switzerland. Lago di Origlio is located in its territory....
, the golden ratio is the proportion between the central section and the side sections of the house.

Painting

Leonardo da Vinci

Leonardo di ser Piero da Vinci was an Italy polymath, being a scientist, mathematician, engineer, inventor, anatomist, Painting, sculptor, architect, botanist, musician and writer....
's illustrations in De Divina Proportione (On the Divine Proportion) and his views that some bodily proportions exhibit the golden ratio have led some scholars to speculate that he incorporated the golden ratio in his own paintings. Some suggest that his Mona Lisa
Mona Lisa

Mona Lisa is a 16th century portrait painting painted in oil painting on a poplar panel painting by Leonardo da Vinci during the Italian Renaissance....
, for example, employs the golden ratio in its geometric equivalents. Whether Leonardo proportioned his paintings according to the golden ratio has been the subject of intense debate. The secretive Leonardo seldom disclosed the bases of his art, and retrospective analysis of the proportions in his paintings can never be conclusive.

Salvador Dalí

Salvador Dalí

Salvador Domingo Felipe Jacinto Dal? i Dom?nech, 1st Marquis of P?bol was a Spain Catalonia surrealist painter born in Figueres.Dal? was a skilled Technical drawing, best known for the striking and bizarre images in his surrealism work....
explicitly used the golden ratio in his masterpiece, The Sacrament of the Last Supper
The Sacrament of the Last Supper

Completed in 1955 after nine months of work, Salvador Dal?s painting The Sacrament of the Last Supper has remained one of his most popular compositions....
. The dimensions of the canvas are a golden rectangle. A huge dodecahedron, with edges in golden ratio to one another, is suspended above and behind Jesus and dominates the composition.

Mondrian

Piet Mondrian

Pieter Cornelis Mondriaan, after 1912 Mondrian, , was a Dutch people Painting.He was an important contributor to the De Stijl art movement and group, which was founded by Theo van Doesburg....
used the golden section extensively in his geometrical paintings.

A statistical study on 565 works of art of different great painters, performed in 1999, found that these artists had not used the golden ratio in the size of their canvases. The study concluded that the average ratio of the two sides of the paintings studied is 1.34, with averages for individual artists ranging from 1.04 (Goya) to 1.46 (Bellini). On the other hand, Pablo Tosto listed over 350 works by well-known artists, including more than 100 which have canvasses with golden rectangle and root-5 proportions, and others with proportions like root-2, 3, 4, and 6.

Book design

According to Jan Tschichold, "There was a time when deviations from the truly beautiful page proportions 2:3, 1:v3, and the Golden Section were rare. Many books produced between 1550 and 1770 show these proportions exactly, to within half a millimetre."

Perceptual studies

Studies by psychologists, starting with Fechner, have been devised to test the idea that the golden ratio plays a role in human perception of beauty

Beauty

Beauty is a characteristic of a person, Location , Object , or idea that provides a perception experience of pleasure, Value , or satisfaction....
. While Fechner found a preference for rectangle ratios centered on the golden ratio, later attempts to carefully test such a hypothesis have been, at best, inconclusive.

Music

James Tenney

James Tenney was an United States composer and influential music theory....
reconceived his piece For Ann (rising)
For Ann (rising)

For Ann is a piece of electronic music created by James Tenney in 1969.Tenney is the author of Meta Hodos, one of, if not the, earliest applications of Gestalt psychology and cognitive science to music, and later "Hierarchical temporal gestalt perception in music: a metric space model" with Larry Polansky, and other works, the influen...
, which consists of up to twelve computer-generated upwardly glissando

Glissando

A glissando is a glide from one pitch to another. It is an Italianized Musical terminology derived from the French glisser, to glide....
ing tones (see Shepard tone

Shepard tone

A Shepard tone, named after Roger Shepard, is a sound consisting of a superposition of sine waves separated by octaves. When played with the base Pitch of the tone moving upwards or downwards, it is referred to as the Shepard scale....
), as having each tone start so it is the golden ratio (in between an equal tempered minor

Minor sixth

A minor sixth is the smaller of two commonly occurring musical intervals that span six diatonic scale degrees. The prefix 'minor' identifies it as being the smaller of the two ; its larger counterpart being a major sixth....
and major sixth

Major sixth

A major sixth is the larger of two commonly occurring musical intervals that span six diatonic scale degrees. The prefix 'major' identifies it as being the larger of the two ; its smaller counterpart being a minor sixth....
) below the previous tone, so that the combination tones produced by all consecutive tones are a lower or higher pitch already, or soon to be, produced.

Erno Lendvai

Erno Lendvai

Erno Lendvai was one of the first theorists to write on the appearance of the golden section and Fibonacci series and how these are implemented in B?la Bart?k's music....
analyzes Béla Bartók

Béla Bartók

B?la Viktor J?nos Bart?k was a Hungarian people composer and pianist, considered to be one of the greatest composers of the 20th century. Through his collection and analytical study of folk music, he was one of the founders of ethnomusicology....
's works as being based on two opposing systems, that of the golden ratio and the acoustic scale

Acoustic scale

In music, the acoustic scale is a seven note Scale which, starting on C, contains the notes: C, D, E, F, G, A and B. This differs from the major scale in having a sharp fourth and flat seventh scale degree....
, though other music scholars reject that analysis. In Bartok's Music for Strings, Percussion and Celesta
Music for Strings, Percussion and Celesta

Music for Strings, Percussion and Celesta Sz. 106, BB 114 is one of the best-known Musical composition by the Hungary composer B?la Bart?k. Commissioned by Paul Sacher to celebrate the tenth anniversary of the Basel Chamber Orchestra, the score is dated September 7, 1936....
the xylophone progression occurs at the intervals 1:2:3:5:8:5:3:2:1. French composer Erik Satie

Erik Satie

Alfred ?ric Leslie Satie was a France composer and pianist. Starting with his first composition in 1884, he signed his name as Erik Satie....
used the golden ratio in several of his pieces, including Sonneries de la Rose+Croix.

The golden ratio is also apparent in the organisation of the sections in the music of Debussy's Image, Reflections in Water, in which "the sequence of keys is marked out by the intervals 34, 21, 13 and 8, and the main climax sits at the phi position."

The musicologist Roy Howat has observed that the formal boundaries of La Mer

La Mer (Debussy)

La mer, trois esquisses symphoniques pour orchestre , or simply La Mer , is an orchestral musical composition by the France impressionist composer Claude Debussy....
correspond exactly to the golden section. Trezise finds the intrinsic evidence "remarkable," but cautions that no written or reported evidence suggests that Debussy consciously sought such proportions. Also, many works of Chopin, mainly Etudes (studies) and Nocturnes, are formally based on the golden ratio. This results in the biggest climax of both musical expression and technical difficulty after about 2/3 of the piece.

Pearl Drums

Pearl Drums

The is a multinational corporation based in Japan with a wide range of products, predominately percussion instruments and flutes. It was founded in 1952....
positions the air vents on its Masters Premium models based on the golden ratio. The company claims that this arrangement improves bass response and has applied for a patent

Patent

A patent is a set of exclusive rights granted by a state to an inventor or his assignee for a term of patent in exchange for a disclosure of an invention....
on this innovation.

In the opinion of author Leon Harkleroad, "Some of the most misguided attempts to link music and mathematics have involved Fibonacci numbers and the related golden ratio."

Nature

Adolf Zeising

Adolf Zeising was a Germany psychologist, whose main interests were mathematics and philosophy.Among his discoveries, Zeising found the golden ratio expressed in the arrangement of branches along the stems of plants and of veins in leaves....
, whose main interests were mathematics and philosophy, found the golden ratio expressed in the arrangement of branches along the stems of plants and of veins in leaves. He extended his research to the skeletons of animals and the branchings of their veins and nerves, to the proportions of chemical compounds and the geometry of crystals, even to the use of proportion in artistic endeavors. In these phenomena he saw the golden ratio operating as a universal law. Zeising wrote in 1854:

Mathematics

Golden ratio conjugate

The negative root of the quadratic equation for f (the "conjugate root") is 1 - ? ˜ -0.618. The absolute value of this quantity (˜ 0.618) corresponds to the length ratio taken in reverse order (shorter segment length over longer segment length, b / a), and is sometimes referred to as the golden ratio conjugate. It is denoted here by the capital Phi (F):

Alternatively, F can be expressed as

This illustrates the unique property of the golden ratio among positive numbers, that

or its inverse:

Short proofs of irrationality

Contradiction from an expression in lowest terms

Recall that:

the whole is the longer part plus the shorter part;

the whole is to the longer part as the longer part is to the shorter part.

If we call the whole n and the longer part m, then the second statement above becomes

n is to m as m is to n − m,

or, algebraically

To say that f is rational means that f is a fraction n/m where n and m are integers. We may take n/m to be in lowest terms and n and m to be positive. But if n/m is in lowest terms, then the identity labeled (*) above says m/(n − m) is in still lower terms. That is a contradiction that follows from the assumption that f is rational.

Derivation from irrationality of v5

Another short proof—perhaps more commonly known—of the irrationality of the golden ratio makes use of the closure

Closure (mathematics)

In mathematics, a Set is said to be closed under some operation if the Operation on members of the set produces a member of the set. For example, the real numbers are closed under subtraction, but the natural numbers are not: 3 and 7 are both natural numbers, but the result of 3 − 7 is not....
of rational numbers under addition and multiplication. If is rational, then is also rational, which is a contradiction if it is already known that the square root of a non-square

Square number

In mathematics, a square number, sometimes also called a perfect square, is an integer that can be written as the square of some other integer; in other words, it is the product of some integer with itself....
natural number

Natural number

In mathematics, a natural number can mean either an element of the Set = *n = = ? = ? ...
is irrational.

Alternate forms

The formula ? = 1 + 1/? can be expanded recursively to obtain a continued fraction

Continued fraction

In mathematics, a continued fraction is an expression such aswhere a0 is an integer and all the other numbers ai are positive integers....
for the golden ratio:

and its reciprocal:

The convergent

Convergent (continued fraction)

A convergent is one of a sequence of values obtained by evaluating successive truncations of a continued fraction. The nth convergent is also known as the nth approximant of a continued fraction....
s of these continued fractions (1, 2, 3/2, 5/3, 8/5, 13/8, … , or 1, 1/2, 2/3, 3/5, 5/8, 8/13, …) are ratios of successive Fibonacci numbers.

The equation ?² = 1 + ? likewise produces the continued square root

Square root

In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x....
form:

Also:

These correspond to the fact that the length of the diagonal of a regular pentagon is f times the length of its side, and similar relations in a pentagram

Pentagram

A pentagram is the shape of a five-pointed star drawn with five straight strokes. The word pentagram comes from the Greek language word pe?t???a???? , a noun form of pe?t???a???? or pe?t???a???? , a word meaning roughly "five-lined" or "five lines"....
.

If x agrees with ? to n decimal places, then agrees with it to 2n decimal places.

An equation derived in 1994 connects the golden ratio to the Number of the Beast

Number of the Beast

The Number of the Beast is a concept from the Book of Revelation of the New Testament of the Christian Bible. The number is 666 in most manuscripts of the New Testament, and in modern translations and Textual criticism....
(666): Which can be combined into the expression: This relationship depends upon the choice of the degree

Degree (angle)

A degree , usually denoted by ? , is a measurement of plane angle, representing 1/360 of a Turn ; one degree is equivalent to p/180 radians....
as the measure of angle, and will not hold when using other units of angular measure.

Geometry

The number f turns up frequently in geometry

Geometry

Symmetry

Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance; such that it reflects beauty or perfection....
. The length of a regular pentagon

Pentagon

In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540?....
's diagonal

Diagonal

A diagonal can refer to a line joining two nonconsecutive vertices of a polygon or polyhedron, or in informal contexts any upward or downward sloping line....
is f times its side. The vertices of a regular icosahedron

Icosahedron

In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....
are those of three mutually orthogonal golden rectangle

Golden rectangle

Algorithm

In mathematics, computing, linguistics and related subjects, an algorithm is a sequence of finite instructions, often used for calculation and data processing....
to arrange a given number of nodes evenly on a sphere, for any of several definitions of even distribution (see, for example, Thomson problem
Thomson problem

The Thomson problem is that of determining the minimum energy configuration of N classical electrons on the surface of a sphere . The electrons repel each other with a force given by Coulomb's law....
). However, a useful approximation results from dividing the sphere into parallel bands of equal area and placing one node in each band at longitudes spaced by a golden section of the circle, i.e. 360°/f 222.5°. This method was used to arrange the 1500 mirrors of the student-participatory satellite Starshine-3

STARSHINE

The STARSHINE series of three artificial satellites were student participatory missions sponsored by the United States Naval Research Laboratory ....
..

Golden triangle, pentagon and pentagram

Golden triangle

The golden triangle can be characterised as an isosceles triangle ABC with the property that bisecting

Bisection

In geometry, bisection is the division of something into two equal or congruent parts, usually by a line , which is then called a bisector. The most often considered types of bisectors are segment bisectors and angle bisectors....
the angle C produces a new triangle CXB which is a similar triangle to the original.

If angle BCX = a, then XCA = a because of the bisection, and CAB = a because of the similar triangles; ABC = 2a from the original isosceles symmetry, and BXC = 2a by similarity. The angles in a triangle add up to 180°, so 5a = 180, giving a = 36°. So the angles of the golden triangle are thus 36°-72°-72°. The angles of the remaining obtuse isosceles triangle AXC (sometimes called the golden gnomon

Golden triangle (mathematics)

A golden triangle is an isosceles triangle triangle in which the two longer sides have equal lengths and in which the ratio of this length to that of the third, smaller side is the golden ratio....
) are 36°-36°-108°.

Suppose XB has length 1, and we call BC length f. Because of the isosceles triangles BC=XC and XC=XA, so these are also length f. Length AC = AB, therefore equals f+1. But triangle ABC is similar to triangle CXB, so AC/BC = BC/BX, and so AC also equals f². Thus f² = f+1, confirming that f is indeed the golden ratio.

Pentagram

The golden ratio plays an important role in regular pentagons and pentagrams. Each intersection of edges sections other edges in the golden ratio. Also, the ratio of the length of the shorter segment to the segment bounded by the 2 intersecting edges (a side of the pentagon in the pentagram's center) is f, as the four-color illustration shows.

The pentagram includes ten isosceles triangles: five acute and five obtuse isosceles triangles. In all of them, the ratio of the longer side to the shorter side is f. The acute triangles are golden triangle

Golden triangle (mathematics)

Ptolemy's theorem

The golden ratio can also be confirmed by applying Ptolemy's theorem

Ptolemy's theorem

Ptolemy's theorem is a relation in Euclidean geometry between the four sides and two diagonals of a cyclic quadrilateral . The theorem is named after the Roman Greece astronomy and mathematics Ptolemy ....
to the quadrilateral formed by removing one vertex from a regular pentagon. If the quadrilateral's long edge and diagonals are b, and short edges are a, then Ptolemy's theorem gives b² = a² + ab which yields

Scalenity of triangles

Consider a triangle

Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....
with sides of lengths a, b, and c in decreasing order. Define the "scalenity" of the triangle to be the smaller of the two ratios a/b and b/c. The scalenity is always less than f and can be made as close as desired to f.

Relationship to Fibonacci sequence

The mathematics of the golden ratio and of the Fibonacci sequence

Fibonacci number

In mathematics, the Fibonacci numbers are a sequence of numbers named after Leonardo of Pisa, known as Fibonacci . Fibonacci's 1202 book Liber Abaci introduced the sequence to Western European mathematics, although the sequence had been previously described in Indian mathematics....
are intimately interconnected. The Fibonacci sequence is:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, …

The closed-form expression

Closed-form expression

In mathematics, an expression is said to be a closed-form expression if, and only if, it can be expressed analytically in terms of a bounded number of certain "well-known" function s....
(known as Binet

Jacques Philippe Marie Binet

Jacques Philippe Marie Binet was a France mathematician, physicist and astronomer born in Rennes; he died in Paris, France, in 1856. He made significant contributions to number theory, and the mathematical foundations of matrix algebra which would later lead to important contributions by Cayley and others....
's formula, even though it was already known by Abraham de Moivre

Abraham de Moivre

Abraham de Moivre was a France mathematician famous for de Moivre's formula, which links complex numbers and trigonometry, and for his work on the normal distribution and probability theory....
) for the Fibonacci sequence involves the golden ratio:

The golden ratio is the limit

Limit (mathematics)

In mathematics, the concept of a "limit" is used to describe the behavior of a Function as its argument or input either "gets close" to some point, or as the argument becomes arbitrarily large; or the behavior of a sequence's elements as their index increases indefinitely....
of the ratios of successive terms of the Fibonacci sequence (or any Fibonacci-like sequence):

Therefore, if a Fibonacci number is divided by its immediate predecessor in the sequence, the quotient approximates f; e.g., 987/610 ˜ 1.6180327868852. These approximations are alternately lower and higher than f, and converge on f as the Fibonacci numbers increase, and:

Furthermore, the successive powers of f obey the Fibonacci recurrence

Recurrence

Recurrence and recurrent may refer to:*Recurrence relation, an equation which defines a sequence recursively*Poincar? recurrence theorem, Henri Poincar?'s theorem on dynamical systems...
:

This identity allows any polynomial in f to be reduced to a linear expression. For example:

Other properties

The golden ratio has the simplest expression (and slowest convergence) as a continued fraction expansion of any irrational number (see Alternate forms above). It is, for that reason, one of the worst cases

Continued fraction

In mathematics, a continued fraction is an expression such aswhere a0 is an integer and all the other numbers ai are positive integers....
of the Lagrange's approximation theorem. This may be the reason angles close to the golden ratio often show up in phyllotaxis

Phyllotaxis

In botany, phyllotaxis or phyllotaxy is the arrangement of the leaf on the plant stem of a plant....
(the growth of plants).

The defining quadratic polynomial and the conjugate relationship lead to decimal values that have their fractional part in common with f:

The sequence of powers of f contains these values 0.618…, 1.0, 1.618…, 2.618…; more generally, any power of f is equal to the sum of the two immediately preceding powers:

As a result, one can easily decompose any power of f into a multiple of f and a constant. The multiple and the constant are always adjacent Fibonacci numbers. This leads to another property of the positive powers of f:

If , then:

When the golden ratio is used as the base of a numeral system

Numeral system

A numeral system is a writing system for expressing numerals , and a mathematical notation for representing numbers of a given set, using graphemes or symbols in a consistent manner....
(see Golden ratio base

Golden ratio base

Golden ratio base is a Non-standard positional numeral systems that uses the golden ratio as its base . It is sometimes referred to as base-f, golden mean base, phi-base, or, colloquially, phinary....
, sometimes dubbed phinary or f-nary), every integer has a terminating representation, despite f being irrational, but every fraction has a non-terminating representation.

The golden ratio is the fundamental unit of the algebraic number field

Algebraic number field

In mathematics, an algebraic number field F is a finite, field extension of the field of rational numbers Q. Thus F is a field that contains Q and has finite Hamel dimension, when considered as a vector space over Q....
and is a Pisot-Vijayaraghavan number

Pisot-Vijayaraghavan number

In mathematics, a Pisot-Vijayaraghavan number, also called simply a Pisot number or a PV number, is an algebraic integer α which is real and exceeds 1, but such that its conjugate elements are all less than 1 in absolute value or modulus....
.

Also,

Decimal expansion

The golden ratio's decimal expansion can be calculated directly from the expression

with v5 ˜ 2.2360679774997896964. The square root of 5 can be calculated with the Babylonian method

Methods of computing square roots

This article presents and explains several methods which can be used to calculate square roots....
, starting with an initial estimate such as x1 = 2 and iterating

Iterative method

In computational mathematics, an iterative method attempts to solve a problem by finding successive approximations to the solution starting from an initial guess....

for n = 1, 2, 3, …, until the difference between x_n and x_n-1 becomes zero, to the desired number of digits.

The Babylonian algorithm for v5 is equivalent to Newton's method

Newton's method

In numerical analysis, Newton's method is perhaps the best known method for finding successively better approximations to the zeroes of a Real number-valued function ....
for solving the equation x² - 5 = 0. In its more general form, Newton's method can be applied directly to any algebraic equation

Algebraic equation

In mathematics, an algebraic equation over a given Field is an equation of the formwhere P and Q are polynomials over that field. For example...
, including the equation x² - x - 1 = 0 that defines the golden ratio. This gives an iteration that converges to the golden ratio itself,

for an appropriate initial estimate x1 such as x1 = 1. A slightly faster method is to rewrite the equation as x - 1 - 1/x = 0, in which case the Newton iteration becomes

These iterations all converge quadratically; that is, each step roughly doubles the number of correct digits. The golden ratio is therefore relatively easy to compute with arbitrary precision

Arbitrary-precision arithmetic

In computer science, arbitrary-precision arithmetic, also called bignum arithmetic, is a technique whereby computer programs perform calculations on integers or rational numbers with an arbitrary number of numerical digits of precision , typically limited only by the available memory of the host system....
. The time needed to compute n digits of the golden ratio is proportional to the time needed to divide two n-digit numbers. This is considerably faster than known algorithms for the transcendental number

Transcendental number

In mathematics, a transcendental number is a number that is not algebraic number, that is, not a solution of a non-zero polynomial equation with rational number coefficients....
s p

Pi or p is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in Euclidean geometry; this is the same value as the ratio of a circle's area to the square of its radius....
and e

E (mathematical constant)

The mathematical constant e is the unique real number such that the function ex has the same value as the derivative, for all values of x....
.

An easily programmed alternative using only integer arithmetic is to calculate two large consecutive Fibonacci numbers and divide them. The ratio of Fibonacci numbers F₂₅₀₀₁ and F₂₅₀₀₀, each over 5000 digits, yields over 10,000 significant digits of the golden ratio.

Millions of digits of f are available . See the web page of Alexis Irlande for the 17,000,000,000 first digits.

Pyramids

Both Egyptian pyramids and those mathematical regular square pyramid

Square pyramid

In geometry, a square pyramid is a Pyramid having a square base. If the apex is perpendicularly above the center of the square, it will have C4v symmetry....
s that resemble them can be analyzed with respect to the golden ratio and other ratios.

Mathematical pyramids and triangles

A pyramid in which the apothem (slant height along the bisector of a face) is equal to f times the semi-base (half the base width) is sometimes called a golden pyramid. The isosceles triangle that is the face of such a pyramid can be constructed from the two halves of a diagonally split golden rectangle (of size semi-base by apothem), joining the medium-length edges to make the apothem. The height of this pyramid is times the semi-base (that is, the slope of the face is ); the square of the height is equal to the area of a face, f times the square of the semi-base.

The medial right triangle of this "golden" pyramid (see diagram), with sides is interesting in its own right, demonstrating via the Pythagorean theorem

Pythagorean theorem

In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a triangle#Types of triangles....
the relationship or . This "Kepler triangle

Kepler triangle

A Kepler triangle is a Special right triangle with edge lengths in geometric progression. The ratio of the edges of a Kepler triangle are linked to the golden ratio...
" is the only right triangle proportion with edge lengths in geometric progression

Geometric progression

In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio....
, just as the 3–4–5 triangle is the only right triangle proportion with edge lengths in arithmetic progression

Arithmetic progression

In mathematics, an arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant....
. The angle with tangent corresponds to the angle that the side of the pyramid makes with respect to the ground, 51.827… degrees (51° 49' 38").

A nearly similar pyramid shape, but with rational proportions, is described in the Rhind Mathematical Papyrus

Rhind Mathematical Papyrus

The Rhind Mathematical Papyrus , is named after Alexander Henry Rhind, a Scotland antiquarian, who purchased the papyrus in 1858 in Luxor, Egypt; it was apparently found during illegal excavations in or near the Ramesseum....
(the source of a large part of modern knowledge of ancient Egyptian mathematics

Egyptian mathematics

Egyptian mathematics refers to the style and methods of mathematics performed in Ancient Egypt....
), based on the 3:4:5 triangle; the face slope corresponding to the angle with tangent 4/3 is 53.13 degrees (53 degrees and 8 minutes). The slant height or apothem is 5/3 or 1.666… times the semi-base. The Rhind papyrus has another pyramid problem as well, again with rational slope (expressed as run over rise). Egyptian mathematics did not include the notion of irrational numbers, and the rational inverse slope (run/rise, multiplied by a factor of 7 to convert to their conventional units of palms per cubit) was used in the building of pyramids.

Another mathematical pyramid with proportions almost identical to the "golden" one is the one with perimeter equal to 2p times the height, or h:b = 4:p. This triangle has a face angle of 51.854° (51°51'), very close to the 51.827° of the Kepler triangle

Kepler triangle

A Kepler triangle is a Special right triangle with edge lengths in geometric progression. The ratio of the edges of a Kepler triangle are linked to the golden ratio...
. This pyramid relationship corresponds to the coincidental relationship .

Egyptian pyramids very close in proportion to these mathematical pyramids are known.

Egyptian pyramids

In the mid nineteenth century, Röber studied various Egyptian pyramids including Khafre, Menkaure and some of the Gizeh, Sakkara and Abusir groups, and was interpreted as saying that half the base of the side of the pyramid is the middle mean of the side, forming what other authors identified as the Kepler triangle; many other mathematical theories of the shape of the pyramids have also been explored.

One Egyptian pyramid is remarkably close to a "golden pyramid" – the Great Pyramid of Giza

Great Pyramid of Giza

The Great Pyramid of Giza, also called Khufu's Pyramid or the Pyramid of Khufu, and Pyramid of Cheops, is the oldest and largest of the three Egyptian pyramidss in the Giza Necropolis bordering what is now Cairo , Egypt, and is the only remaining member of the Seven Wonders of the Ancient World....
(also known as the Pyramid of Cheops or Khufu). Its slope of 51° 52' is extremely close to the "golden" pyramid inclination of 51° 50' and the p-based pyramid inclination of 51° 51'; other pyramids at Giza (Chephren, 52° 20', and Mycerinus, 50° 47') are also quite close. Whether the relationship to the golden ratio in these pyramids is by design or by accident remains controversial. Several other Egyptian pyramids are very close to the rational 3:4:5 shape.

Adding fuel to controversy over the architectural authorship of the Great Pyramid, Eric Temple Bell

Eric Temple Bell

Eric Temple Bell was a mathematician and science fiction author born in Scotland who lived in the United States for most of his life. He published his non-fiction under his given name and his fiction as John Taine....
, mathematician and historian, claimed in 1950 that Egyptian mathematics would not have supported the ability to calculate the slant height of the pyramids, or the ratio to the height, except in the case of the 3:4:5 pyramid, since the 3:4:5 triangle was the only right triangle known to the Egyptians and they did not know the Pythagorean theorem nor any way to reason about irrationals such as p or f.

Michael Rice asserts that principal authorities on the history of Egyptian architecture have argued that the Egyptians were well acquainted with the golden ratio and that it is part of mathematics of the Pyramids, citing Giedon (1957). Historians of science have always debated whether the Egyptians had any such knowledge or not, contending rather that its appearance in an Egyptian building is the result of chance.

In 1859, the pyramidologist

Pyramidology

Pyramidology is a term used, sometimes disparagingly, to refer to various pseudoscience speculations regarding pyramids, most often the Great Pyramid of Giza in Egypt....
John Taylor claimed that, in the Great Pyramid of Giza

Great Pyramid of Giza

or is a letter derived from the Latin alphabet. Both glyphs of the majuscule and Lower case forms of this letter are based on the rotated form of a minuscule e; a similar letter with identical minuscule is used in the Pan-Nigerian Alphabet, but has the capital form majuscule , based on a horizontally flipped majuscule E....
to the ground, to half the length of the side of the square base, equivalent to the secant

Secant

Secant is a term in mathematics. It comes from the Latin secare . It can refer to:* a secant line, in geometry* the Trigonometric functions#Reciprocal functions, reciprocal to the cosine....
of the angle ?. The above two lengths were about 186.4 and 115.2 meters respectively. The ratio of these lengths is the golden ratio, accurate to more digits than either of the original measurements. Similarly, Howard Vyse

Richard William Howard Vyse

Major-General Sir Richard William Howard Vyse was a Kingdom of Great Britain soldier, anthropologist and Egyptologist. He was also Member of Parliament for Beverley and Honiton ....
, according to Matila Ghyka, reported the great pyramid height 148.2 m, and half-base 116.4 m, yielding 1.6189 for the ratio of slant height to half-base, again more accurate than the data variability.

Disputed sightings

Examples of disputed observations of the golden ratio include the following:

Historian John Man states that the pages of the Gutenberg Bible
Gutenberg Bible

The Gutenberg Bible is a printed version of the Latin Vulgate translation of the Bible that was printed by Johannes Gutenberg, in Mainz, Germany in the fifteenth century....
were "based on the golden section shape". However, according to Man's own measurements, the ratio of height to width was 1.45.
In 1991, Jean-Claude Perez proposed a connection between DNA
DNA

Deoxyribonucleic acid is a nucleic acid that contains the genetics instructions used in the development and functioning of all known living organisms and some viruses....
base sequences and gene sequences and the golden ratio. Another such connection, between the Fibonacci numbers and golden ratio and Chargaff's second rule
Chargaff's rules

Chargaff's rules state that DNA from any cell of all organisms should have a 1:1 ratio of pyrimidine and purine bases and, more specifically, that the amount of guanine is equal to cytosine and the amount of adenine is equal to thymine....
concerning the proportions of nucleobase
Nucleobase

Nucleobases are the parts of DNA and RNA that may be involved in pairing . The main ones are cytosine, guanine, adenine , thymine and uracil , abbreviated as C, G, A, T, and U, respectively....
s in the human genome
Human genome

The human genome is the genome of Homo sapiens, which is stored on 23 chromosome pairs. Twenty-two of these are autosome, while the remaining pair is XY sex-determination system....
, was proposed in 2007.
*Australia
Australia

Australia, officially the Commonwealth of Australia, is a country in the southern hemisphere comprising the Australia of the world's smallest continent, the major island of Tasmania, and numerous list of islands of Australia in the Indian Ocean and Pacific Oceans....
n sculptor Andrew Rogers
Andrew Rogers (sculptor)

Andrew Rogers is one of Australia's most distinguished contemporary abstract bronze sculptors with an international reputation. He exhibits internationally and his sculptures are in numerous private and prominent public collections in Australia, S.E....
's 50-ton stone and gold sculpture entitled Ratio, installed outdoors in Jerusalem
Jerusalem

Jerusalem is the capital of Israel and its List of Israeli cities in both population and area, with a population of 747,600 residents over an area of if Positions on Jerusalem East Jerusalem is included....
. Despite the sculpture's sometimes being referred to as "Golden Ratio," it is not proportioned according to the golden ratio, and the sculptor does not call it that: the height of each stack of stones, beginning from either end and moving toward the center, is the beginning of the Fibonacci sequence: 1, 1, 2, 3, 5, 8. His sculpture Ascend in Sri Lanka, also in his Rhythms of Life series, is similarly constructed, with heights 1, 1, 2, 3, 5, 8, 13, but no descending side.
It is sometimes claimed that the number of bees in a beehive divided by the number of drones yields the golden ratio. In reality, the proportion of drones in a beehive varies greatly by beehive, by bee race, by season, and by beehive health status; normal hive populations range from 5,000 to 20,000 bees, while drone numbers range "from none in the winter to as many as 1,500 in the spring and summer" (Graham, 1992, pp 350), thus the ratio is normally much greater than the golden ratio. * This misunderstanding may arise because in theory bees have approximately this ratio of male to female ancestors (See The Bee Ancestry Code
Fibonacci number

In mathematics, the Fibonacci numbers are a sequence of numbers named after Leonardo of Pisa, known as Fibonacci . Fibonacci's 1202 book Liber Abaci introduced the sequence to Western European mathematics, although the sequence had been previously described in Indian mathematics....
) - the caveat being that ancestry can trace back to the same drone by more than one route, so the actual numbers of bees do not need to match the formula.
Some specific proportions in the bodies of many animals (including humans) and parts of the shells of mollusks and cephalopods are often claimed to be in the golden ratio. There is actually a large variation in the real measures of these elements in specific individuals, and the proportion in question is often significantly different from the golden ratio. The ratio of successive phalangeal bones of the digits and the metacarpal bone has been said to approximate the golden ratio. The Nautilus
Nautilus

Nautilus is the common name of any marine creatures of the cephalopod family Nautilidae, the sole family of the suborder Nautilina....
shell, the construction of which proceeds in a logarithmic spiral, is often cited, usually with the idea that any logarithmic spiral is related to the golden ratio, but sometimes with the claim that each new chamber is proportioned by the golden ratio relative to the previous one; however, measurements of Nautilus shells do not support this claim.
The proportions of different plant components (numbers of leaves to branches, diameters of geometrical figures inside flowers) are often claimed to show the golden ratio proportion in several species. In practice, there are significant variations between individuals, seasonal variations, and age variations in these species. While the golden ratio may be found in some proportions in some individuals at particular times in their life cycles, there is no consistent ratio in their proportions.
In investing, some practitioners of technical analysis
Technical analysis

Technical analysis is a security analysis technique that claims the ability to forecast the future direction of prices through the study of past market data, primarily price and volume....
use the golden ratio to indicate support of a price level, or resistance to price increases, of a stock or commodity; after significant price changes up or down, new support and resistance levels are supposedly found at or near prices related to the starting price via the golden ratio. The use of the golden ratio in investing is also related to more complicated patterns described by Fibonacci numbers; see, e.g. Elliott wave principle
Elliott wave principle

The Elliott wave principle is a form of technical analysis that attempts to forecast market trends in the financial markets and other collective activities....
. However, other market analysts have published analyses suggesting that these percentages and patterns are not supported by the data.

External links

Geometry instruction with problems to solve. Information and activities by a mathematics professor.* by Michael Schreiber, Wolfram Demonstrations Project
Wolfram Demonstrations Project

The Wolfram Demonstrations Project is a website developed by Wolfram Research, whose stated goal is to bring computational exploration to the widest possible audience....
, 2007.

Golden ratio

Calculation

History

Timeline

Aesthetics

Architecture

Painting

Book design

Perceptual studies

Music

Nature

Mathematics

Golden ratio conjugate

Short proofs of irrationality

Contradiction from an expression in lowest terms

Derivation from irrationality of v5

Alternate forms

Geometry

Golden triangle, pentagon and pentagram

Golden triangle

Pentagram

Ptolemy's theorem

Scalenity of triangles

Relationship to Fibonacci sequence

Other properties

Decimal expansion

Pyramids

Mathematical pyramids and triangles

Egyptian pyramids

Disputed sightings

See also

Further reading

External links